# Full text of "History Of The Theory Of Numbers - I"

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```CHAP, viii]            NUMBER OF BOOTS OF CONGRUENCES.                       233
N=N* (mod p), where N*+l is derived from either of Hurwitz's two sums for N+ 1 by replacing p by P. The same replacement in Hurwitz's expression for A — 1 leads to the invariant A*— 1, where A* is congruent modulo p to the number of distinct sets of solutions hi the Galois field of order pnm of the equation /(a?!, z2) -0.
G. Rados47 considered the sets of solutions of
, y) =s        x»-2+a(» **-3+ . . . +a£!a)ir*-a=0 (mod p)
for a prune p. Let Ak denote the matrix of D, in (3), with a{ replaced by a,ik\ Let C denote the determinant of order (p— I)2 obtained from D by replacing ak by matrix Ak. Then/=0 has a solution other than z=y=0 if and only if C is divisible by p; it has exactly r sets of solutions other than x==i/=0 if and only if C is of rank (p— I)2— r.
To obtain theorems including the possible solution rc=2/=0, use
<l>(x, y) = 2 W} *"-1+a{*)o;'>-2+ . . . +a<,*21)2/*-fc-1=0 (mod p),
QI    . . .    ap_3       ap_2       «P-i \ 02    ...    ap_2       Op-i+aoO I          0
and ak derived from a by replacing a* by af\ Let 7 be the determinant derived from |a| by replacing ak by matrix ak and 0 by a matrix whose p2 elements are zeros. Then <£=0 has a set of real solutions if and only if 7=0 (mod p); it has r sets of solutions if and only if 7 is of rank p2—r.
*P. B. Schwacha48 discussed the number of roots of congruences.